Remainder Term Research Articles

Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms

Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms

Asymptotic expansion of solutions to the wave equation with space-dependent damping

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy methods with suitable supersolutions of the corresponding parabolic problem.

Gauss-type quadrature rules for variable-sign weight functions

When the Gauss quadrature formula Gn is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval [a,b]. In the present paper, we introduce a Gauss-type quadrature formula Qn for weight functions that change the sign in the interior of [a,b]. Construction of Qn is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of [a,b] at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of Qn are pairwise distinct and contained in the interior of [a,b]. Moreover, Gn (with a non-negative weight function) turns out to be a special case of Qn. Obtained results on the remainder term of Qn suggest that the application of Qn makes sense both when the points from the interior of [a,b] at which the weight function changes sign are known exactly, as well as when those points are known approximately. The accuracy of Qn is confirmed by numerical examples.

HF trimer: 12D fully coupled quantum calculations of HF-stretch excited intramolecular and intermolecular vibrational states using contracted bases of intramolecular and intermolecular eigenstates.

We present the computational methodology, which for the first time allows rigorous twelve-dimensional (12D) quantum calculations of the coupled intramolecular and intermolecular vibrational states of hydrogen-bonded trimers of flexible diatomic molecules. Its starting point is the approach that we introduced recently for fully coupled 9D quantum calculations of the intermolecular vibrational states of noncovalently bound trimers comprised of diatomics treated as rigid. In this paper, it is extended to include the intramolecular stretching coordinates of the three diatomic monomers. The cornerstone of our 12D methodology is the partitioning of the full vibrational Hamiltonian of the trimer into two reduced-dimension Hamiltonians, one in 9D for the intermolecular degrees of freedom (DOFs) and another in 3D for the intramolecular vibrations of the trimer, and a remainder term. These two Hamiltonians are diagonalized separately, and a fraction of their respective 9D and 3D eigenstates is included in the 12D product contracted basis for both the intra- and intermolecular DOFs, in which the matrix of the full 12D vibrational Hamiltonian of the trimer is diagonalized. This methodology is implemented in the 12D quantum calculations of the coupled intra- and intermolecular vibrational states of the hydrogen-bonded HF trimer on an abinitio calculated potential energy surface (PES). The calculations encompass the one- and two-quanta intramolecular HF-stretch excited vibrational states of the trimer and low-energy intermolecular vibrational states in the intramolecular vibrational manifolds of interest. They reveal several interesting manifestations of significant coupling between the intra- and intermolecular vibrational modes of (HF)3. The 12D calculations also show that the frequencies of the v = 1, 2 HF stretching states of the HF trimer are strongly redshifted in comparison to those of the isolated HF monomer. Moreover, the magnitudes of these trimer redshifts are much larger than that of the redshift for the stretching fundamental of the donor-HF moiety in (HF)2, most likely due to the cooperative hydrogen bonding in (HF)3. The agreement between the 12D results and the limited spectroscopic data for the HF trimer, while satisfactory, leaves room for improvement and points to the need for a more accurate PES.

Incorporating the external zeros and poles of the integrand into Gauss-type quadrature rules

Quadrature formulas are often constructed to be exact on the space of functions that are easily integrated and that are in some sense similar to the integrand. This motivates us to explore how the known properties of the integrand can be used to improve the accuracy of certain quadrature rules. In the present paper, we propose an l-point Gauss-type quadrature rule Gl into which the zeros and poles of the integrand outside the integration interval are incorporated. Formula Gl proves to be exact for certain rational functions which have the same zeros and poles as the integrand. It converges, all its nodes are pairwise distinct and belong to the interior of the integration interval, and all its weights are positive. Theoretical results on the remainder term of Gl suggest that formula Gl is applicable both when the incorporated zeros and poles of the integrand are known exactly, as well as when they are known approximately. To practically and economically estimate the error of Gl, some extensions that inherit the l nodes of Gl are developed. They are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Numerical experiments confirm the accuracy of Gl and its extensions.

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

Sharp Weyl laws with singular potentials

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\'c to the case of Schr\"odinger operators with singular potentials.

On some weighted fourth-order equations

This paper deals with the following radial Caffarelli-Kohn-Nirenberg-type inequality,∫RN|x|α|Δu|2dx≥Srad(N,α)(∫RN|x|l|u|pα⁎dx)2pα⁎,u∈Cc∞(RN), where N≥3, 2<α<N, l=4(α−2)(N−2)N−α−α and pα⁎=2(N+l)N−4+α. Then we consider the related Euler-Lagrange equation:Δ(|x|αΔu)=|x|lupα⁎−1,u>0inRN. For α≠0 or l≠0, it is known the solutions of above equation are invariant for dilations λN−4+α2u(λx) but not for translations. However we show that if α is an even integer, there exist new solutions to the linearized problem, related to the radial solution that “replace” the ones due to the translations invariance. As an application, we turn back to investigate the remainder terms of the above inequality. As well as some fourth-order Liouville-type equations with singular data.

Richter’s local limit theorem, its refinement, and related results*

We give a detailed exposition of the proof of Richter’s local limit theorem in a refined form and establish the stability of the remainder term in this theorem under small perturbations of the underlying distribution (including smoothing).We also discuss related quantitative bounds for characteristic functions and Laplace transforms.

Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function . The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [Ehrenmark U. Summability experiments with a class of divergent inverse Kontorovich-Lebedev transforms. Comput Math Appl. 2018;76(1):141–154.]) whether these integrals converge for even entire functions of the exponential type in a weak sense.

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

Formal asymptotic expansions of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation with weak diffusion and small nonlinearity are constructed in the critical case. Under certain conditions imposed on the data of the problem, an asymptotic expansion of the solution is constructed in the form of series in powers of a small parameter with coefficients depending on stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. It is shown that the leading term of the solution asymptotics is determined by solving Cauchy problems for a parabolic Burgers-type equation and, under certain conditions, for a Korteweg–de Vries–Burgers type equation. The remainder terms are estimated with respect to the residual.

A twist in sharp Sobolev inequalities with lower order remainder terms

Abstract Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B &gt; 0 {K,B&gt;0} such that for any u ∈ H 1 ⁢ ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ⁢ ∥ ∇ A ⁡ u ∥ L 2 2 + B ⁢ ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq K\|\nabla_{A}u\|_{L^{2}}^{2}+B\|u\|_{L^{1}}^{2} where H 1 ⁢ ( M ) {H^{1}(M)} is the standard Sobolev space of functions in L 2 {L^{2}} with one derivative in L 2 {L^{2}} , | ∇ A ⁡ u | 2 = A ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) {|\nabla_{A}u|^{2}=A(\nabla u,\nabla u)} and 2 ⋆ {2^{\star}} is the critical Sobolev exponent for H 1 {H^{1}} . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality.

Electricity price forecast based on the STL-TCN-NBEATS model

Taking long-term high-frequency electricity price data as the research content, this paper proposes seasonal and trend decomposition using loess-temporal convolutional network-neural basis expansion analysis for an interpretable time series forecasting (STL-TCN-NBEATS) model to solve the problems of low forecast accuracy caused by high volatility, high frequency and nonlinearity and poor interpretability of the deep learning model. By comparing the forecast effects of the temporal convolutional network-long short-term memory (TCN-LSTM), LSTM and other models, the main conclusions are as follows: (1) The hybrid model, STL-TCN-NBEATS, selected in this paper can effectively solve the problem of low forecast accuracy after reasonable selection of model parameters. The evaluation indexes of the root mean square error (RMSE) and the mean absolute percentage error (MAPE) were 3.7441 and 4.5044, respectively, which were 3.1416 and 2.1336 lower than those of the second-best model (TCN-LSTM). Compared with the autoregressive integrated moving average model (ARIMA), the accuracy was improved by approximately 49.18% (RMSE) and 60.35% (MAPE). (2) The STL-TCN-NBEATS model has better feature extraction ability, so it can obtain higher forecast accuracy. Since the construction of the TCN introduces extended causal convolution and residual blocks, the deep learning network model has better processing ability and robustness for large sample time series. Moreover, the NBEATS network structure enables the model to be trained quickly, and the experimental results verify the effectiveness and high accuracy of this method. (3) The model not only has high precision but also has some interpretability. By decomposing time series data into a trend term, period term and remainder term, the NBEATS and the TCN are used to process the trend term, period term and remainder term, respectively, so that the hybrid model can forecast electricity prices according to the traditional time series processing mode.

The extremal problem for Sobolev inequalities with upper order remainder terms

The extremal problem for Sobolev inequalities with upper order remainder terms

A note on “Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands” by M. M. Spalević et al.

In paper D. Lj. Đjukić, R. M. Mutavdžić Đjukić, A. V. Pejčev, and M. M. Spalević, Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses – a survey of recent results, Electron. Trans. Numer. Anal., 53 (2020), pp. 352–382, Lemma 4.1 can be applied to show the asymptotic behaviour of the modulus of the complex kernel in the remainder term of all the quadrature formulas in the recent papers that are concerned with error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses. However, in the paper D. R. Jandrlić, Dj. M. Krtinić, Lj. V. Mihić, A. V. Pejčev, M. M. Spalević, Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands, Electron. Trans. Anal. 55 (2022), pp. 424–437, which this note is concerned with, there is a kernel whose numerator contains an infinite series, and in this case the mentioned lemma cannot be applied. This note shows that the modulus of the latter kernel attains its maximum as conjectured in the latter paper.

Approximation by λ-Bernstein type operators on triangular domain

In this paper, a new type of ?-Bernstein operators (Bwm,?g)(w,z) and (Bzn ?g)(w,z), their Products Pmn,?g (w,z), Qnm,?g (w, z), and their Boolean sums Smn,?g (w,z), Tnm,?g (w, z) are constructed on triangle h with parameter ? [1,1]. Convergence theorem for Lipschitz type continuous functions and a Voronovskaja-type asymptotic formula are studied for these operators. Remainder terms for error evaluation by using the modulus of continuity are discussed. Graphical representations are added to demonstrate the consistency of theoretical findings for the operators approximating functions on the triangular domain. Also, we show that the parameter ? will provide flexibility in approximation; in some cases, the approximation will be better than its classical analogue.

Long wavelength limit of non-isentropic Euler-Poisson system

This paper is devoted to study the long wavelength limit of Euler-Poisson system with variable temperature for ions in plasma. By the Gardner-Morikawa transformation, we formally derive the Korteweg-de Vries (KdV) equation and the linear KdV equation from the non-isentropic Euler-Poisson system. Then, we establish the rigorous justification of the KdV equation from Euler-Poisson equation in a time interval $ [0, \tau_*\varepsilon^{-\frac{3}{2}}] $ for some $ \tau_*&gt;0 $. The long wavelength limit of Euler-Poisson equation follows from the suitable uniform bound of the remainder terms by the nonlinear energy estimates.

Green's function and pointwise behaviors of the one-dimensional modified Vlasov-Poisson-Boltzmann system

The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl. Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear mVPB system in terms of the Green's function.

$\varOmega$-result for the remainder term in Beurling’s prime number theorem for well-behaved integers

We obtain a new $\varOmega $-result for the remainder term $\psi (x)-x$ of a Beurling prime system for which the integers are very well-behaved in the sense that $N(x)=ax + \mathrm O(x^\beta )$ for some $a \gt 0$ and $\beta \lt 1/2$. As part

Hyperbolic summation for functions of the GCD and LCM of several integers

Let kge 2 be a fixed integer. We consider sums of type sum _{n_1cdots n_kle x} F(n_1,ldots ,n_k), taken over the hyperbolic region {(n_1,ldots ,n_k)in {mathbb {N}}^k: n_1cdots n_kle x}, where F:{mathbb {N}}^krightarrow {mathbb {C}} is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations sum _{n_1cdots n_kle x} f((n_1,ldots ,n_k)) and sum _{n_1cdots n_kle x} f([n_1,ldots ,n_k]), involving the GCD and LCM of the integers n_1,ldots ,n_k, where f:{mathbb {N}}rightarrow {mathbb {C}} belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k=2.